The authors in [1]} use their assumed upper bound on \(\mathcal {E}_{\delta }(A)\)  to obtain the following lower bound on the multiplicative energy of \(A\) :
\(\sum _{a,b\in A}\#\big ((a.A)_{\delta ^+} \cap (b.A)_{\delta ^+}\big )\gtrsim K_4^{-1}(\#A)^3.\) 
